Angle Bisectors, Reflection and Shortest Paths

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. You and your house are on the same side of a straight river. On the way home, you must first touch the river to fill a bottle. So where should you touch it, to make the total walk the shortest? One small trick, a reflection, answers this exactly. But first, the bisectors of the angles between two lines. Both ideas come up often in JEE coordinate geometry, so let's begin.

0:30Bisectors: equally far from both lines

So, here are two lines, L one and L two; the small one and two are subscripts that number them. The first is a one times x, plus b one times y, plus c one, equals zero, and the second has twos. Okay? Now, a bisector cuts the angle between two lines into two equal halves. Let me ask you this: what is special about any point on a bisector? Yes, exactly! It is at the same distance from both lines. So a bisector is all the points equally far from L one and L two. Clear? From the last concept, distance from a line is a modulus over a square root; the straight bars are the modulus sign. So the first expression's modulus, over root of a one squared plus b one squared, equals the same for the second line. Okay? Now, how do we remove these modulus bars? If two numbers have the same modulus, what can you say about them? That is correct, they are equal, or negatives of each other! So the bars go, and we get the plus or minus sign, a plus written over a minus. Two signs, two bisectors, one for each pair of opposite angles. Got it?

1:46Which sign gives which bisector?

Now, let's sort out which bisector is which. Step one: look at the constants, c one and c two. If their signs differ, multiply one whole equation by minus one. Okay? Then the plus sign gives the bisector of the angle in which the origin lies. Why? At the origin, the expressions are just c one and c two, the same sign, and they keep their signs all through that angle, so its bisector takes the plus sign. Right? So tell me, does the plus sign always give the bisector of the acute angle? Oh no, careful, it does not; this is a common trap. Work out a one times a two, plus b one times b two. If it is negative, the plus bisector is the acute one. With me? And if it is positive, the plus bisector is the obtuse one, so the minus sign gives the acute one. But only after making the constants the same sign. Clear?

2:43Example 1: bisectors of 3x + 4y = 5 and 12x - 5y = 7

Alright, let's try one from the course. Find the bisectors of the angles between three x plus four y equals five, and twelve x minus five y equals seven. Pause the video and try it first. Done? Okay, let's check. Everything to one side: three x plus four y minus five equals zero, and root of three squared plus four squared is five. Okay? Similarly, twelve x minus five y minus seven equals zero, and root of twelve squared plus five squared is thirteen. Right? Into the formula, and cross multiply. Thirteen times, three x plus four y minus five, equals plus or minus, five times, twelve x minus five y minus seven. With the plus sign: thirty nine x plus fifty two y minus sixty five, equals sixty x minus twenty five y minus thirty five. Collect the terms; the double arrow means which gives. Twenty one x minus seventy seven y plus thirty equals zero. With the minus sign, the right side becomes minus sixty x plus twenty five y plus thirty five. Collecting terms gives ninety nine x plus twenty seven y minus one hundred equals zero. Now a bonus, using our rule. Which of these two bisects the acute angle? Let's see. The constants, minus five and minus seven, are both negative, the same sign. And three times twelve, plus four times minus five, is sixteen, which is positive. So the plus bisector is the obtuse one, and the acute one is ninety nine x plus twenty seven y minus one hundred equals zero. Clear?

4:35Reflection and the shortest path

Now, reflection. Here is the last concept's figure: a point P, x one comma y one, the dark line a x plus b y plus c equals zero, and d, the distance from P to it. The orange perpendicular meets the line at M, the foot, at a right angle, the small square. Go as far again, and you reach P prime, the image; the dash is read prime. So P M is perpendicular to the line, the upside down T, and P M equals M P prime. Okay? Now think of the line as a mirror. When a ray of light bounces off it, the angle with the line going in equals the angle coming out. Right? Now our river puzzle, in maths. The arrows mean: from A to a point of the line, then on to B, okay? And A and B are on the same side of the line. Let me ask you this. If B were on the other side of the line, what would the shortest path be? Yes, exactly, just the straight segment from A to B, which already crosses the line. Nothing beats a straight line, right? So here is the trick for the same side: reflect B in the line, to get its image, B prime, on the other side. Join A to B prime; it cuts the line at the point P we want. Now let's see why this is the shortest. The line is the perpendicular bisector of B B prime, so any point on it is equally far from B and B prime. So A P plus P B equals A P plus P B prime, which is greater than or equal to A B prime, equal exactly when P is on A B prime. That's it!

6:16Example 2: least AP + PB for A(1, 2), B(5, 4) via the x-axis

Now, remember our river? Here it is with numbers. The points A, one comma two, and B, five comma four, lie above the x axis. Find the point P on the x axis that makes A P plus P B least, and that least value. Pause the video and try it first. Done? Okay, let's check. Both points are above the axis, the same side, so we reflect B in the axis. Quick question: what is the image of five comma four in the x axis? Yes, exactly! In the x axis, only the sign of y changes. So B prime is five comma minus four, on the other side from A, okay? Now the line A B prime. Its slope is minus four minus two, over five minus one, which is minus six over four, or minus three over two. Through A, the line is y minus two equals minus three over two, times x minus one. Clear? Now we find where it meets the x axis. Put y equals zero: minus two equals minus three over two, times x minus one, which gives x equals seven over three. So P is seven over three, comma zero. And the minimum is just the length of A B prime. Root of four squared plus six squared, that is root of sixteen plus thirty six, which is root of fifty two, or two times root thirteen. River puzzle solved: reflect the house, and walk straight towards its image! Got it?

7:54Quick revision: bisector signs, acute test, shortest path

So, quick revision. Bisectors: one fraction equals plus or minus the other. With constants of the same sign, plus gives the origin's angle, acute when a one times a two, plus b one times b two, is negative. And for the shortest path via a line, same side, reflect B and join A to B prime. Now try the practice problem on this page. All the best!